Simultaneous multi-mode surveillance provides enhanced situational awareness providing several concurrent data products that complement one another. Synthetic aperture radar (SAR) and ground moving target indication (GMTI) are traditionally performed separately, either on separate platforms or on a single platform employing mode switching, as they have differing requirements. Recent efforts in the field have in part concentrated on exploiting a common data source for both modalities in order to perform them simultaneously.
To this end, the frequency jump burst (FJB) waveform is a likely candidate for a common SAR/GMTI waveform as the narrowband transmit waveforms provide processing flexibility at the receiver. Each of the subpulses comprising the FJB can be processed independently of the others for GMTI—yielding multiple looks. For SAR, the waveforms must be combined in some fashion to yield the expected range resolution promised by the overall transmit bandwidth. This topic is the main focus of this invention, in which we present a novel method for wideband synthesis (WBS).
In a previous method used by Lord, Davis, et al, an appropriate time delay is applied to each of the received waveforms to synthesize wideband waveforms prior to pulse compression (V. Murthy, U. Pillai and M. E. Davis, “Waveforms for Simultaneous SAR and GMTI,” in Proceedings of the 2012 IEEE Radar Conference, Atlanta, Ga. May 7-11, 2012; M. Davis, “Challenges of Ultra Wideband, Multi-Mode Radar,” 2011 IEEE CIE International Conference on Radar, vol. 1, October 2011, pp. 5-8; R. Lord and M. Inggs, “High Resolution SAR Processing Using Stepped Frequencies,” IEEE IGARSS '97, 1997). Here each (k+1)th subpulse is delayed by kT, where T is the subpulse duration and 0≦k≦K−1. Therefore no time delay is applied to the first subpulse, and all subsequent subpulses are adjusted accordingly (“the first subpulse” refers to the first subpulse in each sequence of K). This method is fairly intuitive, as it mimics construction of a wideband linear frequency modulation (LFM) waveform where each segment contains a different portion of the entire spectrum and increases (or decreases) as a function of time.
Once the wideband waveform is synthesized and the wideband matched filter is generated, SAR imaging can be applied to the pulse compressed data. The position from which the first subpulse was transmitted is used in the SAR imaging algorithm, as all of the other K−1 subpulses are compensated to the position of the first subpulse. This method works well for far ranges, and particularly at broadside and low integration angles.
At either near range, or when the squint angle is significant (greater than three to four degrees,) this method does not work very well. At broadside and far range the differential delays between the subpulses can be written as kT, for the (k+1)th subpulse (V. Murthy, U. Pillai and M. E. Davis, “Waveforms for Simultaneous SAR and GMTI,” in Proceedings of the 2012 IEEE Radar Conference, Atlanta, Ga. May 7-11, 2012; M. Davis, “Challenges of Ultra Wideband, Multi-Mode Radar,” 2011 IEEE CIE International Conference on Radar, vol. 1, October 2011, pp. 5-8; R. Lord and M. Inggs, “High Resolution SAR Processing Using Stepped Frequencies,” IEEE IGARSS '97, 1997). It would appear, from the point spread functions generated for various cases, that this is an approximation to the true differential delay between the subpulses which is accurate under the conditions that the scene being imaged is at far range (>20 km) and at or near broadside. The true parabolic delays can be computed for each pixel imaged, and for every platform position and every sensor/subarray. However this brute-force processing leads to significant redundancy and an extremely high computation cost.
We begin first by establishing the signal model for the LFM base case to which the wideband synthesis method will be compared to. The LFM signal of length KTf(t)=ej2πβt2, 0t KT  (1)is modulated with an RF carrier at frequency f0 and transmitted. β in (1) is the chirp rate and the total bandwidth of the waveform in (1) is given byBW=2βKT.  (2)The transmit signal is given bys(t)=f(t)ej2πf0t.  (3)The receive waveform zx,y(t,m) is a function of both fast time t and pulse index 0≦m≦M−1zx,y(t,m)=αx,ys(tτm(x,y))  (4)where Σm(x,y) represents the two-way delay between the platform and a scatter point located at (x,y) for the mth pulse; αx,y represents the scatterer coefficient for the scatter located at (x,y). The total received data for the LFM waveform is given by
                              z          ⁡                      (                          t              ,              m                        )                          =                                            ∑                              x                ,                y                                      ⁢                                          z                                  x                  ,                  y                                            ⁡                              (                                  t                  ,                  m                                )                                              =                                    ∑                              x                ,                y                                      ⁢                                          α                                  x                  ,                  y                                            ⁢                              s                ⁡                                  (                                                                                    t                                                                                                                          τ                            m                                                    ⁡                                                      (                                                          x                              ,                              y                                                        )                                                                                                                                )                                                                                        (        5        )            where the received signals due to different scatters are summed together. The data is ready to be pulse compressed as using the matched filter h(t)=f*(t). Once pulse compression has been applied asg(t,m)=z(t,m)h(t).  (6)An imaging algorithm can be used to obtain the final SAR map
                              S          ⁢                                          ⁢          A          ⁢                                          ⁢                                    R              LFM                        ⁡                          (                                                x                  ~                                ,                                  y                  ~                                            )                                      =                ⁢                              ∑                          m              =              0                                      M              -              1                                ⁢                                    g              ⁡                              (                                                      t                    +                                                                  τ                        m                                            ⁡                                              (                                                                              x                            ~                                                    ,                                                      y                            ~                                                                          )                                                                              ,                  m                                )                                      ⁢                                                  ⁢                          (              7              )                                                              =                ⁢                              M            ⁢                                                  ⁢                          α                                                x                  ~                                ,                                  y                  ~                                                      ⁢                          s              ⁡                              (                t                )                                      ⁢                                                  ⁢            ◯            *            h            ⁢                          (              t              )                                +                                                ⁢                              (                                          ∑                                  m                  =                  0                                                  M                  -                  1                                            ⁢                                                ∑                                                            x                      ,                      y                      ,                                                                                      (                                                                              x                            ~                                                    ,                                                      y                            ~                                                                          )                                            ≠                                              (                                                  x                          ,                          y                                                )                                                                                            ⁢                                                      α                                          x                      ,                      y                                                        ⁢                                      s                    ⁡                                          (                                              t                        -                                                  [                                                                                                                    τ                                m                                                            ⁡                                                              (                                                                  x                                  ,                                  y                                                                )                                                                                      -                                                                                          τ                                m                                                            ⁡                                                              (                                                                                                      x                                    ~                                                                    ,                                                                      y                                    ~                                                                                                  )                                                                                                              ]                                                                    )                                                                                            )                    ⁢          ◯          *                      h            ⁡                          (              t              )                                          where {tilde over (x)} and {tilde over (y)} are the image coordinates in terms of meters. The SAR image formation in (7) align the return signals due to M pulses according two-way delay between the platform position and the image point ({tilde over (x)},{tilde over (y)}). The SAR imaging algorithm can be the backprojection imaging algorithm (L. A. Gorham and L. J. Moore., “SAR image formation toolbox for MATLAB,” Algorithms for Synthetic Aperture Radar Imagery XVII, 7669, 2010.). Notice that the term Mα{tilde over (x)},{tilde over (y)}s(t)h(t) in (7) represents the mainlobe and it corresponds to the contribution from the scatterer located at the image point ({tilde over (x)},{tilde over (y)}) and the term
      (                  ∑                  m          =          0                          M          -          1                    ⁢                        ∑                                    x              ,              y              ,                                                      (                                                      x                    ~                                    ,                                      y                    ~                                                  )                            ≠                              (                                  x                  ,                  y                                )                                                    ⁢                              α                          x              ,              y                                ⁢                      s            ⁡                          (                              t                -                                  [                                                                                    τ                        m                                            ⁡                                              (                                                  x                          ,                          y                                                )                                                              -                                                                  τ                        m                                            ⁡                                              (                                                                              x                            ~                                                    ,                                                      y                            ~                                                                          )                                                                              ]                                            )                                            )    ⁢  ◯  *      h    ⁡          (      t      )      represents the sidelobe and it corresponds to the contributions from scatterers located at other locations.
The LFM case can be used as a known prior art base case to which one or more embodiments of the present invention will be compared later in this application. Now that we have a base case to compare to, we can look at the frequency jump burst (FJB) waveform. The FJB waveform at intermediate frequency (IF) is given byf(t,k)=ej2πβ(t+kT)2, 0≦t≦T  (8)where β is the chirp rate and 0≦k≦K−1 is the subpulse index over the K subpulses. The bandwidth of each subpulse is BW/K. The transmit signal corresponding to (8) iss(t,k)=f(t,k)ej2πf0(t+kT),  (9)An advantage of writing the FJB waveform in this manner is that the phase between subpulses is continuous and allows us to reconstruct the full wideband LFM in (1)
                                          ∑                          k              =              0                                      K              -              1                                ⁢                      s            ⁡                          (                                                t                  -                  kT                                ,                k                            )                                      =                              s            ⁡                          (              t              )                                .                                    (        10        )            This is an important property that will be used to show the equivalence between WBS and the LFM case. The received signal due to a scatterer located at (x,y) can be written aszx,y(t,m,k)=αx,ys(tτm,k(x,y),k).  (11)In (11) the two way delay term τm,k(x,y) is indexed not only by the pulse number m, but also by the subpulse number k as each of the subpulses are transmitted sequentially. The two-way delay for the FJB can be written as
                                          τ                          m              ,              k                                ⁡                      (                          x              ,              y                        )                          =                              2            c                    ⁢                                                                      x                  2                                -                                                      (                                                                  a                                                  m                          ,                          k                                                                    -                      y                                        )                                    2                                                      .                                              (        12        )            τm,0(x,y) in (12) is same as τm(x,y) in (4),τm,0≡τm.  (13)In (12), (x,y) is a point on the ground. The first FJB subpulse (k=0) is transmitted at the same time/location as each of the LFM pulses, as denoted in (13). The total received data for the (k+1)th FJB transmit waveform is given by
                              z          ⁡                      (                          t              ,              m              ,              k                        )                          =                                            ∑                              x                ,                y                                      ⁢                                          z                                  x                  ,                  y                                            ⁡                              (                                  t                  ,                  m                  ,                  k                                )                                              =                                    ∑                              x                ,                y                                      ⁢                                          α                                  x                  ,                  y                                            ⁢                              s                ⁡                                  (                                                            t                      -                                                                        τ                                                      m                            ,                            k                                                                          ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                                      ,                    k                                    )                                                                                        (        14        )            where the received signals due to different scatters are summed together.
The previous method used by Lord, Davis, et al. (V. Murthy, U. Pillai and M. E. Davis, “Waveforms for Simultaneous SAR and GMTI,” in Proceedings of the 2012 IEEE Radar Conference, Atlanta, Ga. May 7-11, 2012; M. Davis, “Challenges of Ultra Wideband, Multi-Mode Radar,” 2011 IEEE CIE International Conference on Radar, vol. 1, October 2011, pp. 5-8; R. Lord and M. Inggs, “High Resolution SAR Processing Using Stepped Frequencies,” IEEE IGARSS '97, 1997), consists of stitching a wideband signal by applying the appropriate phase and time shifts prior to matched filtering and imaging. Starting with the received signal in (14), we can get the time compensated signal{tilde over (z)}(t,m,k)=z(tkT,m,k).  (15)In (15), the time compensation have aligned every kth subpulse for 1≦k≦K−1 to the first subpulse (k=0) in time. This compensation is true when the target is located at broadside or the distance between the platform and the target is very large. When generating the point spread function (PSF) using this method, the resulting PSF will have the same features as the LFM if is the scatterer point is located on broadside. From (15), a wideband waveform can be constructed by applying a shift of kT and summing
                              z          ⁡                      (                          t              ,              m                        )                          =                                            ∑                              k                =                0                                            K                -                1                                      ⁢                          z              ⁡                              (                                                      t                    -                    kT                                    ,                  m                  ,                  k                                )                                              =                                    ∑                              k                =                0                                            K                -                1                                      ⁢                                          ∑                                  x                  ,                  y                                            ⁢                                                α                                      x                    ,                    y                                                  ⁢                                                      s                    ⁡                                          (                                                                        t                          -                                                                                    τ                                                              m                                ,                                k                                                                                      ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                -                          kT                                                ,                        k                                            )                                                        .                                                                                        (        16        )            Notice that when the target is at broadside, we haveτm,k(x,y)≃τm(x,y).  (17)Thus, (16) can be written as
                              z          ⁡                      (                          t              ,              m                        )                          =                                            ∑                              k                =                0                                            K                -                1                                      ⁢                          z              ⁡                              (                                                      t                    -                    kT                                    ,                  m                  ,                  k                                )                                              =                                    ∑                              x                ,                y                                      ⁢                                          α                                  x                  ,                  y                                            ⁢                                                s                  ⁡                                      (                                          t                      -                                                                        τ                          m                                                ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                                      )                                                  .                                                                        (        18        )            Here we have used the property in (10) to show that (16) is equivalent to (5).Since (18) is equivalent to (5), the range compressed data can be written asg(t,m)=z(t,m)h(t),  (19)and as with the LFM, the corresponding SAR map can be obtained as
                              S          ⁢                                          ⁢          A          ⁢                                          ⁢                                    R              1                        ⁡                          (                                                x                  ~                                ,                                  y                  ~                                            )                                      =                                            ∑                              m                =                0                                            M                -                1                                      ⁢                          g              ⁡                              (                                                      t                    +                                                                  τ                        m                                            ⁡                                              (                                                                              x                            ~                                                    ,                                                      y                            ~                                                                          )                                                                              ,                  m                                )                                              =                      S            ⁢                                                  ⁢            A            ⁢                                                  ⁢                                          R                LFM                            .                                                          (        20        )            However, when the target is not on the broadside, the synthesized wideband waveform in (16) is in the form of
                              z          ⁡                      (                          t              ,              m                        )                          =                                            ∑                              k                =                0                                            K                -                1                                      ⁢                          z              ⁡                              (                                                      t                    -                    kT                                    ,                  m                  ,                  k                                )                                              =                                    ∑                              k                =                0                                            K                -                1                                      ⁢                                          ∑                                  x                  ,                  y                                            ⁢                                                α                                      x                    ,                    y                                                  ⁢                                                      s                    ⁡                                          (                                                                        t                          -                                                                                    τ                                                              m                                ,                                k                                                                                      ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                -                          kT                                                ,                        k                                            )                                                        .                                                                                        (        21        )            After range compression, the SAR image we obtain is in the form of
                              S          ⁢                                          ⁢          A          ⁢                                          ⁢                                    R              1                        ⁡                          (                                                x                  ~                                ,                                  y                  ~                                            )                                      =                                            ∑                              m                =                0                                            M                -                1                                      ⁢                          g              ⁡                              (                                                      t                    +                                                                  τ                        m                                            ⁡                                              (                                                                              x                            ~                                                    ,                                                      y                            ~                                                                          )                                                                              ,                  m                                )                                              =                                    ∑                              m                =                0                                            M                -                1                                      ⁢                                          ∑                                  k                  =                  0                                                  K                  -                  1                                            ⁢                                                ∑                                      x                    ,                    y                                                  ⁢                                                      α                                          x                      ,                      y                                                        ⁢                                                            s                      ⁡                                              (                                                                              t                            -                                                          [                                                                                                                                    τ                                                                          m                                      ,                                      k                                                                                                        ⁡                                                                      (                                                                          x                                      ,                                      y                                                                        )                                                                                                  -                                                                                                      τ                                    m                                                                    ⁡                                                                      (                                                                                                                  x                                        ~                                                                            ,                                                                              y                                        ~                                                                                                              )                                                                                                                              ]                                                        -                            kT                                                    ,                          k                                                )                                                              .                                                                                                          (        22        )            Notice that in (22), even the scatterer located at ({tilde over (x)},{tilde over (y)}) can not be coherently combined over the M pulses.